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The space-time fractional diffusion equation with Caputo derivatives

  • F. Huang
  • F. Liu
Article

Abstract

We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from standard diffusion equation by replacing the second-order space derivative with a Caputo (or Riemann-Liouville) derivative of order β∈(0, 2] and the first-order time derivative with Caputo derivative of order α∈(0, 1]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We derive explicit expression of the Green function. The Green function also can be interpreted as a spatial probability density function evolving in time. We further explain the similarity property by discussing the scale-invariance of the space-time fractional diffusion equation.

AMS Mathematics Subject Classification

26A33 49K20 44A10 

Key words and phrases

Fractional diffusion equation time-space Caputo derivative Fourier transform Laplace transform Mittag-Leffler function Green function stable probability distributions 

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Copyright information

© Korean Society for Computational and Applied Mathematics 2005

Authors and Affiliations

  1. 1.Department of Mathematical SciencesXiamen UniversityXiamenChina
  2. 2.School of Mathematical SciencesQueensland University of TechnologyAustralia
  3. 3.School of Mathematical SciencesSouth China University of TechnologyGuangzhouChina

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